**Many manufacturers have quality control programs** that include inspection of incoming materials for defects. Suppose a computer manufacturer receives circuit boards in batches of five. Two boards are selected from each batch for inspection. We can represent possible outcomes of the selection process by pairs. For example, the pair (1, 2) represents the selection of boards 1 and 2 for inspection.

a. List the ten different possible outcomes.

b. Suppose that boards 1 and 2 are the only defective boards in a batch. Two boards are to be chosen at random. Define X to be the number of defective boards observed among those inspected. Find the probability distribution of X.

c. Let F(x) denote the cdf of X. First determine $F(0) =P(X\leq 0)$,$F(1)$,$F(2)$ then $F(x)$ for all other x.

### Solution

a. The ten different possible outcomes are as follows

` $$ \begin{aligned} S=& \{ (1,2), (1,3),(1,4),(1,5)\\ & (2,3), (2,4), (2,5)\\ & (3,4),(3,5)\\ & (4,5)\} \end{aligned} $$ `

b. Suppose that boards 1 and 2 are the only defective boards in a batch. Two boards are to be chosen atrandom.

Let $X$ the number of defective boards observed among those inspected.

The probability distribution of X.

` $$ \begin{aligned} P(X=x)= \left\{ \begin{array}{ll} 0.3, & \hbox{$x=0$;} \\ 0.6, & \hbox{$x=1$;} \\ 0.1, & \hbox{$x=2$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{aligned} $$ `

c. Distribution function of $X$ is

$$F(x) = P(X\leq x)=\sum_{x\leq x} P(X=x)$$.

$F(0)= P(X\leq 0)=0.3$,

` $$ \begin{aligned} F(1) &= P(X\leq 1) \\ &=P(X=0)+P(X=1) \\ &=0.3+0.6=0.9 \end{aligned} $$ `

` $$ \begin{aligned} F(2) &= P(X\leq 2)\\ &=P(X=0)+P(X=1)+P(X=2)\\ &=0.3+0.6+0.1\\ &= 1 \end{aligned} $$ `

Therefore,

` $$ \begin{aligned} F(x)= \left\{ \begin{array}{ll} 0, & \hbox{$x<0$;} \\ 0.3, & \hbox{$ x\leq 0$;} \\ 0.9, & \hbox{$x\leq 1$;} \\ 1, & \hbox{$x\geq 2$.} \end{array} \right. \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators